At this point I think I've done something wrong because there is no reference angle for cos(x) = 1/3.

Why not? 1/3 is between -1 and 1, of course, there is a reference angle. Any calculator will tell you that cos(70.53)= 1/3 (approximately). The problem did say "use exact solutions whenever possible.
Or, if you prefer, exact solutions would be and .

2cos(x) + 1 = 0
2cos(x) = -1
cos(x) = -1/2

Quadrants II and III
Reference Angle= 60 degrees

180 - 60 = 120 degrees
180 + 60 = 240 degrees

Therefore, so far, I have solved for the values: 120 degrees and 240 degrees